In a Gaussian mechanism, let \(Y\)  be the random variable following Gaussian distribution with \(Y\sim \mathcal {N}\left(0, \sigma ^2I_d\right)\)  and \(f: X^{n} \rightarrow \mathbb {R}^{d}\) .
The Gaussian mechanism \( M(X)=f(X)+ Y\)  follows the DP profile [1]}:
\(\delta (\epsilon ; \mu )=\Phi \left(-\frac{\epsilon }{\mu }+\frac{\mu }{2}\right)-\mathrm {e}^{\epsilon } \Phi \left(-\frac{\epsilon }{\mu }-\frac{\mu }{2}\right),\) 
